This is an example of a non-simplicial $\mathbb R$-tree (from Wikipedia): Start with the interval $[0,2]$, for each positive integer $n$, glue an interval of length $1$ to the point $1-1/n$ in the original interval. The set of singular points (points whose complement has three or more connected components) is discrete, but fails to be closed since $1$ is an ordinary point in this $\mathbb R$-tree. So gluing an interval to $1$ would result in a closed set of singular points which makes the set of singular points non-discrete. (We call this construction $X$).
And a real tree $T$ is simplicial if and only if the set of singular points of $T$ is discrete in $X$. So the example above is not a simplicial complex.
But $X$ is the set of simplices satisfying (1) every face of a simplex is also in $X$ (trivially true); (2) the non-empty intersection of any simplifies is a face of both of them (trivially true). So isn’t $X$ a simplicial complex from the definition? I wonder where I made wrong to get this contradiction?